It is tempting to mimic Kontsevich’s construction as follows: Let upper C$C$ be a nodal projective connected curve; then a morphism upper C right-arrow script upper M$C \to {\mathcal{M}}$ is said to be a stable map of degree d$d$ if the associated morphism to the coarse moduli scheme upper C right-arrow bold upper M$C \to {\mathbf{M}}$ is a stable map of degree d$d$.

It follows from our results below that the category of stable maps into script upper M${\mathcal{M}}$ is a Deligne-Mumford stack. A somewhat surprising point is that it is not complete.

1.4. Twisted stable maps

Our main goal here is to correct this deficiency. In order to do so, we will enlarge the category of stable maps into script upper M${\mathcal{M}}$. The source curve script upper C${{\mathcal{C}}}$ of a new stable map script upper C right-arrow script upper M${{\mathcal{C}}} \to {\mathcal{M}}$ will acquire an orbispace structure at its nodes. Specifically, we endow it with the structure of a Deligne-Mumford stack.

The case where script upper M${\mathcal{M}}$ is the classifying stack of a finite group upper G$G$ allows one to improve on the spaces of admissible covers, give moduli compactifications of spaces of curves with abelian and nonabelian level structures, and, with a suitable choice of the group upper G$G$, show that there is a smooth, fine moduli space for admissible upper G$G$-covers, which is a finite covering of script upper M overbar Subscript g$\overline {{\mathcal{M}}}_g$. This is the subject of our preprint Referencenormal alef$\aleph$-C-V with Alessio Corti, with some ideas contributed by Johan de Jong. Some of these applications were indicated in our announcement Referencenormal alef$\aleph$-V1. This approach to admissible covers is closely related to the work of Wewers ReferenceWe.

(3)

A similar reasoning applies to curves with r$r$-spin structures, e.g. theta characteristics. This is studied in Referencenormal alef$\aleph$-J.

(4)

The recursive nature of the theorem allows one to construct both minimal models and stable reduction for pluri-fibered varieties. This is related to recent work of Mochizuki ReferenceMo and deserves further study.

(5)

In ReferenceC-R, W. Chen and Y. Ruan introduce Gromov-Witten invariants of an orbifold, using a differential geometric counterpart of our stack of twisted stable maps. In our joint work Referencenormal alef$\aleph$-G-V with Tom Graber we give an algebraic treatment of these Gromov-Witten invariants, and in the special case of 3-pointed genus 0 maps of degree 0, construct the Chen-Ruan product with integer coefficients. See also ReferenceF-G for an algebraic treatment of global quotients.

(6)

In this paper we verify that script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$ is a proper stack by going through the conditions one by one. It may be worthwhile to develop a theory of Grothendieck Quot-stacks and deduce our results from such a theory. It seems likely that some of our methods could be useful for developing such a theory.

While our paper was circulating, we were told in 1999 by Maxim Kontsevich that he had also discovered the stack of twisted stable maps, but had not written down the theory. His motivation was in the direction of Gromov-Witten invariants of stacks.

1.6. Acknowledgments

We would like to thank Kai Behrend, Larry Breen, Barbara Fantechi, Ofer Gabber, Johan de Jong, Maxim Kontsevich, and Rahul Pandharipande, for helpful discussions. We are grateful to Laurent Moret-Bailly for providing us with a preprint of the book ReferenceL-MB before it appeared. The first author thanks the Max Planck Institute für Mathematik in Bonn for a visiting period which helped in putting this paper together.

2. Generalities on stacks

2.1. Criteria for a Deligne-Mumford stack

We refer the reader to ReferenceAr and ReferenceL-MB for a general discussion of algebraic stacks (generalizing ReferenceD-M), and to the appendix in ReferenceVi for an introduction. We spell out the conditions here, as we follow them closely in the paper. We are given a category script upper X${\mathcal{X}}$ along with a functor script upper X right-arrow script upper S c h slash double-struck upper S${\mathcal{X}}\to {\mathcal{S}}ch/{\mathbb{S}}$. We assume

These last two conditions are often the most difficult to verify. For the last one, M. Artin has devised a set of criteria for constructing upper X right-arrow script upper X$X\to {\mathcal{X}}$ by algebraization of formal deformation spaces (see ReferenceAr, Corollary 5.2). Thus, in case double-struck upper S${\mathbb{S}}$ is of finite type over a field or an excellent Dedekind domain, condition (3b) holds if

For the notion of properness of an algebraic stack see ReferenceL-MB, Chapter 7. Thus a stack script upper X right-arrow double-struck upper S${\mathcal{X}}\to {\mathbb{S}}$ is proper if it is separated, of finite type and universally closed. In ReferenceL-MB, Remarque 7.11.2, it is noted that the weak valuative criterion for properness using traits might be insufficient for properness. However, in case script upper X${\mathcal{X}}$ has finite diagonal, it is shown in ReferenceE-H-K-V, Theorem 2.7, that there exists a finite surjective morphism from a scheme upper Y right-arrow script upper X$Y \to {\mathcal{X}}$. In such a case the usual weak valuative criterion suffices (ReferenceL-MB, Proposition 7.12).

Recall that an algebraic space bold upper X${\mathbf{X}}$ along with a morphism pi colon script upper X right-arrow bold upper X$\pi \colon {\mathcal{X}}\to {\mathbf{X}}$ satisfying properties (2) and (3) is called a coarse moduli space (or just moduli space). In particular, the theorem of Keel and Mori shows that coarse moduli spaces of algebraic stacks with finite diagonal exist. Moreover, from (4) and (5) above we have that the formation of a coarse moduli space behaves well under flat base change:

Definition 2.3.2

An action of a finite group normal upper Gamma$\Gamma$ on a scheme upper V$V$ is said to be tame if for any geometric point s colon upper S p e c normal upper Omega right-arrow upper V$s\colon \operatorname {Spec}\Omega \to V$, the group upper S t a b left-parenthesis s right-parenthesis$\operatorname {Stab}(s)$ has order prime to the characteristic of normal upper Omega$\Omega$.

The reader can verify that a separated Deligne-Mumford stack is tame if and only if the actions of the groups normal upper Gamma Subscript alpha$\Gamma _\alpha$ on upper V Subscript alpha$V_\alpha$ in Lemma 2.2.3 are tame.

Proof.

By the descent axiom for script upper M${\mathcal{M}}$ (see 2.1 (2)) the problem is local in the étale topology, so we may replace upper X$X$ and bold upper M${\mathbf{M}}$ with the spectra of their strict henselizations at a geometric point; then we can also assume that we have a universal deformation space upper V right-arrow script upper M$V\to {\mathcal{M}}$ which is finite. Now upper U$U$ is the complement of the closed point, upper U$U$ maps to script upper M${\mathcal{M}}$, and the pullback of upper V$V$ to upper U$U$ is finite and étale, so it has a section, because upper U$U$ is simply connected; consider the corresponding map upper U right-arrow upper V$U\to V$. Let upper Y$Y$ be the scheme-theoretic closure of the graph of this map in upper X times Subscript bold upper M Baseline upper V$X\times _{\mathbf{M}}V$. Then upper Y right-arrow upper X$Y\to X$ is finite and is an isomorphism on upper U$U$. Since upper X$X$ satisfies upper S 2$S_2$, the morphism upper Y right-arrow upper X$Y\to X$ is an isomorphism.

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Remark 2.4.2

The reader can verify that the statement and proof work in higher dimension. See also related lemmas in ReferenceMo.

Proof of the lemma.

First note that if upper R Superscript s h$R^{\operatorname {sh}}$ is the strict henselization of upper R$R$, the condition on the action of normal upper Gamma$\Gamma$ allows one to lift it to upper R Superscript s h$R^{\operatorname {sh}}$. Also, the statement that we are trying to prove is local in the étale topology, so by standard limit arguments we can assume that upper R$R$ is strictly henselian. Replacing bold upper M${\mathbf{M}}$ by the spectrum of the strict henselization of its local ring at the image of the closed point of upper R$R$, we can assume that script upper M${{\mathcal{M}}}$ is of the form left-bracket upper V slash upper H right-bracket$[V/H]$, where upper V$V$ is a scheme and upper H$H$ is a finite group. Then the object eta$\eta$ corresponds to a principal upper H$H$-bundleupper P right-arrow upper U$P \to U$, on which normal upper Gamma$\Gamma$ acts compatibly with the action of normal upper Gamma$\Gamma$ on upper U$U$, and an upper H$H$-equivariant and normal upper Gamma$\Gamma$-invariant morphism upper P right-arrow upper V$P \to V$. Since upper U$U$ is strictly henselian, the bundle upper P right-arrow upper U$P \to U$ is trivial, so upper P$P$ is a disjoint union of copies of upper S p e c upper R$\operatorname {Spec}R$, and the group normal upper Gamma$\Gamma$ permutes these copies; furthermore the hypothesis on the action of normal upper Gamma$\Gamma$ on the closed fiber over the residue field insures that normal upper Gamma$\Gamma$ sends each component into itself. The thesis follows easily.

If more than one curve is considered, we will often use the notation normal upper Sigma Subscript i Superscript upper U$\Sigma _i^U$ to specify the curve upper U$U$. On the other hand, we will often omit the subschemes normal upper Sigma Subscript i Superscript upper U$\Sigma _i^U$ from the notation left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Superscript upper U Baseline right-parenthesis$(U \to S, \Sigma _i^U)$ if there is no risk of confusion.

Definition 3.1.6

Let left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis$(U \to S, \Sigma _i)$ be an n$n$-marked curve, with an action of a finite group normal upper Gamma$\Gamma$, and let script upper M${\mathcal{M}}$ be a Deligne-Mumford stack. Given eta element-of script upper M left-parenthesis upper U right-parenthesis$\eta \in {\mathcal{M}}(U)$, an essential action of normal upper Gamma$\Gamma$ on left-parenthesis eta comma upper U right-parenthesis$(\eta , U)$ is a pair of compatible actions of normal upper Gamma$\Gamma$ on eta$\eta$ and on left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis$(U\to S, \Sigma _i)$, with the property that if g$g$ is an element of normal upper Gamma$\Gamma$ different from the identity and u 0$u_0$ is a geometric point of upper U$U$ fixed by g$g$, then the automorphism of the pullback of eta$\eta$ to u 0$u_0$ induced by g$g$ is not trivial.

Furthermore, if s 0$s_0$ is a geometric point of upper S$S$ and u 0$u_0$ a nodal point of the fiber upper U Subscript s 0$U_{s_0}$ of upper U$U$ over s 0$s_0$, then

(1)

the stabilizer normal upper Gamma prime$\Gamma '$ of u 0$u_0$ is a cyclic group which sends each of the branches of upper U Subscript s 0$U_{s_0}$ to itself;

(2)

if k$k$ is the order of normal upper Gamma prime$\Gamma '$, then a generator of normal upper Gamma prime$\Gamma '$ acts on the tangent space of each branch by multiplication with a primitive k$k$-th root of 1$1$.

In particular, each nodal point of upper U Subscript s 0$U_{s_0}$ is sent to a nodal point of upper C Subscript s 0$C_{s_0}$.

Proof.

The first statement follows from the definition of an essential action and the invariance of the arrow eta vertical-bar Subscript upper U Sub Subscript gen Subscript Baseline upper E$\eta |_{U_{\text{gen}}} \to E$.

As for (1), observe that if the stabilizer normal upper Gamma prime$\Gamma '$ of u 0$u_0$ did not preserve the branches of upper U Subscript s 0$U_{s_0}$, then the quotient upper U Subscript s 0 Baseline slash normal upper Gamma prime$U_{s_0}/\Gamma '$, which is étale at the point u 0$u_0$ over the fiber upper C Subscript s 0$C_{s_0}$, would be smooth over upper S$S$ at u 0$u_0$, so u 0$u_0$ would be in the inverse image of upper C Subscript gen$C_{\text{gen}}$. From the first part of the proposition it would follow that normal upper Gamma prime$\Gamma '$ is trivial, a contradiction.

So normal upper Gamma prime$\Gamma '$ acts on each of the two branches individually. The action on each branch must be faithful because it is free on the complement of the set of nodes; this means that the representation of normal upper Gamma prime$\Gamma '$ in each of the tangent spaces to the branches is faithful, and this implies the final statement.

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Definition 3.2.4

A chart is called balanced if for any nodal point of any geometric fiber of upper U$U$, the two roots of 1 describing the action of a generator of the stabilizer on the tangent spaces to each branch of upper U$U$ are inverse to each other.

Proposition 4.1.1

Proof.

First of all let us show that upper C$C$ is flat over upper S$S$. We may assume that upper S$S$ is affine; let upper R$R$ be its coordinate ring. Fix a geometric point c 0 right-arrow upper C$c_0 \to C$, and call upper C Superscript s h$C^{\operatorname {sh}}$ the strict henselization of upper C$C$ at c 0$c_0$. Let upper U$U$ be an étale cover of script upper C${{\mathcal{C}}}$, and u 0$u_0$ a geometric point of upper U$U$ lying over c 0$c_0$; denote by upper U Superscript s h$U^{\operatorname {sh}}$ the strict henselization of upper U$U$ at u 0$u_0$. If normal upper Gamma$\Gamma$ is the automorphism group of the object of script upper C${{\mathcal{C}}}$ corresponding to u 0$u_0$, then normal upper Gamma$\Gamma$ acts on upper U Superscript s h$U^{\operatorname {sh}}$, and upper C Superscript s h$C^{\operatorname {sh}}$ is the quotient upper U Superscript s h Baseline slash normal upper Gamma$U^{\operatorname {sh}}/ \Gamma$. Since script upper C${\mathcal{C}}$ is tame, the order of normal upper Gamma$\Gamma$ is prime to the residue characteristic of u 0$u_0$, therefore the coordinate ring of upper C Superscript s h$C^{\operatorname {sh}}$ is a direct summand, as an upper R$R$-module, of the coordinate ring of upper U Superscript s h$U^{\operatorname {sh}}$, so it is flat over upper R$R$.

The fact that the fibers are nodal follows from the fact that, over an algebraically closed field, the quotient of a nodal curve by a group action is again a nodal curve. Properness is clear; the fact that the morphism script upper C right-arrow upper C${{\mathcal{C}}} \to C$ is surjective implies that the fibers are geometrically connected. The fact that upper C$C$ is of finite presentation is an easy consequence of the fact that script upper C${\mathcal{C}}$ is of finite presentation.

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Following tradition, when we speak of a “family over upper S$S$” or “curve over upper S$S$”, it is always assumed to be of finite presentation.